Legendre's three-square theorem
E620662
Legendre's three-square theorem is a result in number theory that characterizes exactly which positive integers can be expressed as the sum of three squares of integers.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Legendre's three-square theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6800991 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Legendre's three-square theorem Context triple: [Lagrange's four-square theorem, relatedTo, Legendre's three-square theorem]
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A.
Lagrange's four-square theorem
Lagrange's four-square theorem is a fundamental result in number theory stating that every natural number can be expressed as the sum of four integer squares.
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B.
Jacobi’s four-square theorem
Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
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C.
Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
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D.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
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E.
Waring's problem
Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Legendre's three-square theorem Target entity description: Legendre's three-square theorem is a result in number theory that characterizes exactly which positive integers can be expressed as the sum of three squares of integers.
-
A.
Lagrange's four-square theorem
Lagrange's four-square theorem is a fundamental result in number theory stating that every natural number can be expressed as the sum of four integer squares.
-
B.
Jacobi’s four-square theorem
Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
-
C.
Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
-
D.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
-
E.
Waring's problem
Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
result in number theory
ⓘ
theorem ⓘ |
| allowsConclusion | every positive integer not of the form 4^a(8b+7) is a sum of three squares ⓘ |
| appearsIn | classical textbooks on number theory ⓘ |
| appliesTo | all positive integers n ⓘ |
| assumes | x, y, z are integers ⓘ |
| characterizes | which positive integers are sums of three squares ⓘ |
| classificationRole | classifies positive integers by representability as x^2 + y^2 + z^2 ⓘ |
| concerns |
representation of integers as sums of squares
ⓘ
sum of three squares ⓘ |
| domain | positive integers ⓘ |
| excludes | integers of the form 4^a(8b+7) from being sums of three squares ⓘ |
| excludesForm | 4^a(8b+7) ⓘ |
| field | number theory ⓘ |
| formalizedIn | modern proof assistants and formal verification systems ⓘ |
| givesCondition | n is a sum of three squares if and only if n is not of the form 4^a(8b+7) ⓘ |
| givesNecessaryAndSufficientConditionFor | an integer to be a sum of three squares ⓘ |
| hasExample |
1 = 1^2 + 0^2 + 0^2 is a sum of three squares
ⓘ
15 is not a sum of three squares ⓘ 2 = 1^2 + 1^2 + 0^2 is a sum of three squares ⓘ 23 is not a sum of three squares ⓘ 28 = 3^2 + 3^2 + 2^2 is a sum of three squares ⓘ 3 = 1^2 + 1^2 + 1^2 is a sum of three squares ⓘ 31 is not a sum of three squares ⓘ 7 is not a sum of three squares ⓘ |
| hasGeneralization |
local-global criteria for quadratic forms in three variables
ⓘ
results on sums of k squares for k ≥ 3 ⓘ |
| historicalAttribution |
proved by Adrien-Marie Legendre
ⓘ
linked to:
Adrien-Marie Legendre
|
| implies |
if v_2(n) is even and n/4^{v_2(n)} ≡ 7 (mod 8) then n is not a sum of three squares
ⓘ
integers congruent to 7 modulo 8 are not sums of three squares ⓘ |
| namedAfter | Adrien-Marie Legendre ⓘ |
| quantifiesOver | nonnegative integers a and b ⓘ |
| refinedBy | Gauss's work on ternary quadratic forms ⓘ |
| relatedTo |
Lagrange's four-square theorem
ⓘ
Waring's problem ⓘ local-global principle ⓘ sum of two squares theorem ⓘ
linked to:
Fermat's theorem on sums of two squares
ternary quadratic forms ⓘ |
| states | A positive integer n can be written as x^2 + y^2 + z^2 with x,y,z in Z iff n is not of the form 4^a(8b+7) ⓘ |
| timePeriod | 18th century ⓘ |
| typeOfCondition | congruence condition on integers ⓘ |
| usedIn |
additive number theory
ⓘ
classification of representations by ternary quadratic forms ⓘ computational number theory algorithms for sums of squares ⓘ studies of universal quadratic forms ⓘ |
| usesConcept |
modular arithmetic
ⓘ
p-adic methods ⓘ quadratic forms ⓘ square numbers ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Legendre's three-square theorem Description of subject: Legendre's three-square theorem is a result in number theory that characterizes exactly which positive integers can be expressed as the sum of three squares of integers.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.